0. Code

>

restart;

>	with(plots):

Warning, the name changecoords has been redefined

>	ccc := 'color = COLOUR(RGB, .6, .5, .5), thickness = 2':

>

LongDiv  :=  proc( P, r)      local A, C, d, i,j,k,cols,rows,q,L2,dg, Q,R;     d := degree( P);    cols :=  d + 3; rows := 3*d + 3;       for k from 0 to d do     C||k := coeff(P, x, k) ;  od;     A := array( [seq( [  seq(` `, j = 1..cols ) ],   i = 1..rows) ]);     A[3,1] := x-r;     for k from 0 to d do  A[3,d - k + 3] :=  C||k*x^k ; od;     for k from 0 to d do  A[2,d - k + 3] :=  `__`; od;     A[k,2] :=  `|`;     for k from 1 to d do       q := simplify(A[3*k,2+k]/x);  A[1,2+k] := q;       A[1+3*k,1] := A[3,1]* q;   A[1+3*k,2] := ` = `;       L2 := expand(A[3,1]*q);   dg := degree(L2);       A[1+3*k,2+k] := coeff(L2, x, dg)   *x^dg;          A[1+3*k,3+k] := coeff(L2, x, dg-1) *x^(dg-1);       if (k > 1) then A[3*k,3+k] := A[3,3+k];  fi;       for j from k+2 to k+3 do  A[2+3*k,j] :=  `__`; od;       A[3 + 3*k,2+k] := A[3*k,2+k] - A[1+3*k,2+k];          A[3 + 3*k,3+k] := A[3*k,3+k] - A[1+3*k,3+k];     od;     print(A);     Q := quo(  P , (x-r), x, 'R'):     print(` `);print(` `);     print(P/(x+r) = Q + R/(x+r));print(` `);     end proc:

>	PolyLinFact  :=  proc(S,T)     local Q,R;     Q := quo( S, T, x, 'R'):     print( expand('S'/T) = T*(sort(Q,x) + R/T));     end proc:

>

SynDiv  :=  proc( P, r)     local A, C, d, i,j,k;     d := degree( P);     for k from 0 to d do   C||k := coeff(P, x, k) ;  od;     A := array( [seq( [  seq(` `, j = 0..(d+2) ) ],   i = 1..4) ]);     A[1,1] := r;     for k from 0 to d do  A[1,d - k + 3] :=  C||k ; od;     for k from 0 to d do  A[2,d - k + 3] :=  0; od;     for k from 0 to d do  A[3,d - k + 3] :=  `__`; od;     for k from 0 to d do          A[4,k + 3] := A[1,k + 3] + A[2,k + 3];        if(k<d) then A[2,k + 4] := r * A[4,k + 3]; fi         od;     print(A);     end proc:

>

>	N := 4*x^3 + x^2 - 20*x + 3;

>	N := (x-3)*(x+5)*(x-4);

>	PolyLinFact( (x-3)*(x+5)*(x-4), x - 3); PolyLinFact( (x-3)*(x+5)*(x-4), (x - 3)*(x-4));

>	SynDiv( N, +4 );

>	SynDiv( x^4 + x^3 + x^2 + x + 1, -1);

  1. The Remainder Theorem

One version of the Remainder Theorem  is this :

      For a polynomial f(x), the following are equivalent :
        - f(r) = R
        - the remainder when (x-r) is divided into f(x) is R

It makes a connection between the remainder of a polynomial division and evaluating a polynomial. Let's look at some examples.

Example 1.1 :  Let's create a polynomial and pick a number r. Then we will evaluate both of the expressions above and note that the two are the same.

>	f := x -> 3*x^2 - 7*x + 11; r := 4;

>

f(r);

>	LongDiv( f(x), r);

Example 1.2 :   Here is another example.

>	f := x -> x^3 + 10*x^2 - 100*x + 300; r := -1;

>

f(r);

>	LongDiv( f(x), r);

Example 1.3 :  Here is a harder example.

>	f := x -> x^7 - x^6 + 4*x^5 - 9*x^4 + 11*x^3 -2*x^2 + 8*x^2 - 12*x + 144; r := -7/2;

>

f(r);

>	LongDiv( f(x), r);

In all these cases, you can see that the number gotten from evaluating the polynomial is the same as the remainder.
 

  2. Evaluating a Polynomial using Synthetic Division

One application is to evaluate a polynomial by finding the remainder. In some cases, synthetic division is
faster than the actual evaluation.

Example 2.1 :   It might be debatable as to which is easier.

>	f := x -> x^4 + x^3 + x^2 + x + 1;

>

f(-5);

>	SynDiv( f(x), -5 );

Example 2.2 :   Here is another example. Which method is easier and faster?

>	f := x -> x^6 - 170*x^4 + 200*x^2 - 400*x  + 11;

>

f(13);

>	SynDiv( f(x), 13 );

Example 2.3 :   Here is another example.

>	f := x -> x^11 + 3*x^10   - 7* x^9 +  20*x^8 + 30*x^7 + 31*x^6           + 34*x^5  - 110*x^3  - 61*x^2 - 58*x  - 11;

>

f(-5);

>	SynDiv( f(x), -5 );

  3. Finding A Remainder by Evaluating a Polynomial

We can use the theorem in the other direction too. In some cases, its easier to evaluate a polynomials than it is to do a synthetic division.
Example 3.1 :   Find the remainder when  is divided by (x+2).

>	f := x -> x^4 + x^3 + x^2 + x  + 1;

>

f(-2);

>	SynDiv( f(x), -2 );

Example 3.2 :   Find the remainder when  is divided by (x -1).

This synthetic division is a bit large, but its quite easy to simply evaluate the polynomial.

>	f := x -> x^20  +  x^10  + 1;

>

f( 1);

>	SynDiv( f(x),  1 );

Example 3.2 :   Find the remainder when  is divided by (x+1).

Warning: don't try doing the synthetic division unless you have a very large piece of paper!

>	f := x -> x^100 + x^75 + x^50 + x^25  + 1;

>

f(-1);

  4. The Factor Theorem

The Factor Theorem  is a corollary of the Remainder theorem :

      For a polynomial f(x), the following are equivalent :
        - f(r) = 0
        - (x-r) divides f(x)
        - r is a root of f(x)
        - the graph of f(x) has an x-intercept at x = r
     

This theorem makes a connection between the roots of a polynomial, and linear factors. Let's see it in action.

Example 4.1 :  

>	f := x -> 6*x^3+13*x^2-4;

>	r := -2;  f(r); r := 1/2;  f(r); r := -2/3;  f(r);

This polynomial has three roots. We would expect three linear factors .....

>	factor(f(x));

>	plot( f(x), x = -3..2, y = -20..50, ccc );

>	rt := solve( f(x) = 0, x); display( pointplot( {[rt[1],0],[rt[2],0],[rt[3],0]}, symbolsize = 30, color = red),          plot( f(x), x = -3..2, y = -20..50, ccc ));

Example 4.2 :   Here is another example.

>	f := x -> 5*x^3+7*x^2+7*x+2; r := -2/5;

>

f(r);

>	factor(f(x));

>	display( pointplot( [r,0],  symbolsize = 30, color = red),                   plot( f(x), x = -2..1, y = -15..20, ccc ));

Whenever we have a root, we have a linear factor, and thus we have a way of decomposing the polynomial into a product of smaller degreed polynomials.